What is Analytic continuation: Definition and 25 Discussions

In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which it is initially defined becomes divergent.
The step-wise continuation technique may, however, come up against difficulties. These may have an essentially topological nature, leading to inconsistencies (defining more than one value). They may alternatively have to do with the presence of singularities. The case of several complex variables is rather different, since singularities then need not be isolated points, and its investigation was a major reason for the development of sheaf cohomology.

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  1. S

    A Does the Z boson pole show up in the photon propagator?

    If I look at the photon propagator <A_mu (x) A^nu(0) > in momentum space, as I understand it I am to compute this by summing up all the self-energy diagrams of the photon, which look like: photon -> stuff -> photon In particular, since the photon shares the same quantum numbers as the Z, you...
  2. K

    I Pauli-Villars regularization for Vacuum Polarization

    Hello! I am currently reading Itzykson Zuber QFT book and on Chapter 7 where for the first time loops are considered. Particular method of dealing with divergences namely Pauli-Villars regularization is considered in section 7-1-1 considering vacuum polarization diagram. I do understand physics...
  3. A

    I Analytic continuation of Dilogarithm

    I am trying to understand the branching geometry of the Dilogarithm function as described in Branching geometry of Dilogarithm. In Theorem 8.6, the following (dilogarithm) definition is given by letting ##n=2##: $$ \text{Li}_2(z)=\text{Li}_2^{(k_0,k_1)}(z)=\text{Li}_2^{(0)}(z)+\sum_{m=0}^1...
  4. C

    I Analytic continuation of a dilogarithm

    The correct analytical continuation of the dilog function is of the form $$\text{lim}_{\epsilon \rightarrow 0^+} \text{Li}_2(x \pm i\epsilon) = -\left(\text{Li}_2(x) \mp i\pi \ln x \right)$$ I read this in a review at some point which I can no longer find at the moment so just wondered if this...
  5. nomadreid

    I Video (analytic continuation) seems to mix 4-D & 2-D maps

    The question here is not asking for links to help understand analytic continuation or the Riemann hypothesis, but rather help in understand the bits of hand-waving in the following video’s explanations : https://www.youtube.com/watch?v=sD0NjbwqlYw (apparently narrated by the same person who does...
  6. D

    Contour integration & the residue theorem

    When one uses a contour integral to evaluate an integral on the real line, for example \int_{-\infty}^{\infty}\frac{dz}{(1+x)^{3}} is it correct to say that one analytically continues the integrand onto the complex plane and integrate it over a closed contour ##C## (over a semi-circle of radius...
  7. J

    Analytic continuation of Airy function

    For x\in\mathbb{R} we can set \textrm{Ai}(x) = \frac{1}{2\pi} \int\limits_{-\infty}^{\infty} e^{i\big(\frac{t^3}{3} + tx\big)}dt If we substitute in place of x a complex parameter z with \textrm{Im}(z)>0, the integral will converge on [0,\infty[, but diverge on ]-\infty,0]. With...
  8. K

    Analytic continuation and physics

    Analytic continuation can be used in mathematics to assign a finite value to an infinite series that diverges to infinity. Is it correct and legitimate to equate this value to a diverging infinite series that occurs in a physical theory of nature? Will this process give a correct answer that can...
  9. B

    Newest implementation of GW approximation needs analytic continuation?

    I know that at early stage (around 1999), GW implementation uses Matusbara frequency to help calculating self energy, and then apply analytic continuation to change it to real frequency for subsequent calculation. I don't know whether this scheme has been superceded by other implementations for...
  10. alyafey22

    MHB Analytic continuation and Regularization simplified.

    Hello MHB members In this set of lectures we are going to explore the nice idea of analytic continuation and regularization of divergent series and integrals. Don't get panic ,the idea is so simple that you are actually using it without knowing. I'll try to make the tutorials as simple as...
  11. U

    Analytic Continuation along a Loop

    Homework Statement Choose a branch that is analytic in the circle |z-2|<1. Then analytically continue this branch along the curve indicated in Fig 5.18. Do the new functional values agree with the old? a, 3z^{\frac{2}{3}} b, (e^z)^\frac{1}{3} Fig 5.18 is basically an ellipse like loop...
  12. A

    Analytic continuation to find scattering bound states

    Hello, I am trying to understand the idea of using analytic continuation to find bound states in a scattering problem. What do the poles of the reflection coefficent have to do with bound states? In a problem that my quantum professor did in class (from a previous final), we looked at the 1D...
  13. J

    Extending radius of convergence by analytic continuation

    Hi, Suppose I have an analytic function f(z)=\sum_{n=0}^{\infty} a_n z^n the series of which I know converges in at least |z|<R_1, and I have another function g(z) which is analytically continuous with f(z) in |z|<R_2 with R_2>R_1 and the nearest singular point of g(z) is on the circle...
  14. S

    Exploring the Riemann Hypothesis and Analytic Continuation

    I don't know anything of complex analysis or analytic number theory or analytic continuation. But i read about zeta function and riemann hypothesis over wikipedia, clay institute's website and few other sources. I started with original zeta function...
  15. M

    Analytic continuation of an integral involving the mittag-leffler function

    greetings . we have the integral : I(s)=\int_{0}^{\infty}\frac{s(E_{s}(x^{s})-1)-x}{x(e^{x}-1)}dx which is equivalent to =I(s)=\frac{1}{4}\int_{0}^{\infty}\frac{\theta(ix)\left(sE_{s/2} ((\pi x)^{s/2})-s-2x^{1/2}\right)}{x}dx E_{\alpha}(z) being the mittag-leffler function and...
  16. G

    Analytic continuation of the zeta function

    I was reading through the first chapter of Edwards' book on the zeta function, and I'm a little confused about Riemann's original continuation of zeta to all of the complex plane... The zeta function is supposed to be defined for all s in the set of complex numbers by \zeta \left( s \right) =...
  17. C

    How to find analytic continuation?

    The following is the problem from Fetter and Walecka (problem 3.7) If f(z) is defined to be the integration of rho(x) * (z-x)^(-1) from -infinity to +infinity. rho is in the following form rho(x)=gamma * ( gamma^2+x^2 )^(-1). Evaluate f(z) explicitly for Im(z)>0 and find its analytic...
  18. Z

    IR divergences by analytic continuation of parameters

    i had a discussion with a physicist i proposed that in order to avoid the IR divergence \int_{0}^{\infty}dx(x-a)^{-3}x^{2} we could propose as regularized value the value of F(-a) , where F is the integral \int_{0}^{\infty}dx(x+b)^{-3}x^{2} so if we could regularize this simply...
  19. L

    How do you DO analytic continuation?

    they talk about the existence of analytic continuation, but how do you find (the power series/product), calculate, compute the analytic continuation? how do you actually do analytic continuation on a function?
  20. T

    Analytic Continuation: Definition & Uses in QFT

    What is analytic continuation? Seems to be used often in QFT.
  21. benorin

    Exploring Analytic Continuation of Dirichlet Series

    I am writing my senior thesis (I am an undergrad math major at UCSB) on Dirichlet Series, which are, in the classical sense, series of the form \xi (s)=\sum_{n=1}^{\infty}\frac{a_n}{n^s} where a_n,s\in\mathbb{C} and a_n is multiplicative, hence \forall n,m\in\mathbb{N}, \...
  22. S

    Can Analytic Continuation Determine the Metric Over a 4-D Manifold?

    Hello all, Can the method of analytic continuation [1] be applied to the calculation of the metric over a 4-d manifold? In other words, suppose that we are given the value of the metric g_ab as well as its power series at a single point p in a 4-dimensional manifold. Assume further that...
  23. benorin

    Convergence question on analytic continuation of Zeta fcn

    Given \zeta (s) = \sum_{k=1}^{\infty} k^{-s} which converges in the half-plane \Re (s) >1, the usual analytic continuation to the half-plane \Re (s) >0 is found by adding the alternating series \sum_{k=1}^{\infty} (-1)^kk^{-s} to \zeta (s) and simplifing to get \zeta (s) =...
  24. C

    What is analytic continuation?

    I don't understand the concept of analytic continuation, at any level (I have some small amount of experience with introductory undergraduate-level complex analysis from a long time ago, mostly forgotten): 1) Firstly, why would you want to apply analytic continuation on some complex function...